Maclaurin and Taylor Series; Power Series Objective: To take our knowledge of Maclaurin and Taylor polynomials and extend it to series.

Maclaurin and Taylor Series We defined: the nth Maclaurin polynomial for a function as the nth Taylor polynomial for f about x = x 0 as

Maclaurin and Taylor Series It is not a big jump to extend the notions of Maclaurin and Taylor polynomials to series by not stopping the summation index at n. Thus, we have the following definition.

Example 1 Find the Maclaurin series for (a) (b) (c) (d)

Example 1 Find the Maclaurin series for (a) (b) (c) (d) (a) We take the Maclaurin polynomial and extend it.

Example 1 Find the Maclaurin series for (a) (b) (c) (d) (b) We take the Maclaurin polynomial and extend it.

Example 1 Find the Maclaurin series for (a) (b) (c) (d) (c) We take the Maclaurin polynomial and extend it.

Example 1 Find the Maclaurin series for (a) (b) (c) (d) (d) We take the Maclaurin polynomial and extend it.

Example 2 Find the Taylor series for 1/x about x = 1.

Example 2 Find the Taylor series for 1/x about x = 1. We found in the last section that the nth Taylor polynomial for 1/x about x = 1 is

Example 2 Find the Taylor series for 1/x about x = 1. We found in the last section that the nth Taylor polynomial for 1/x about x = 1 is Thus, the Taylor series for 1/x about x = 1 is

Power Series in x Maclaurin and Taylor series differ from the series that we have considered in Sections in that their terms are not merely constants, but instead involve a variable. These are examples of power series, which we now define.

Power Series in x If are constants and x is a variable, then a series of the form is called a power series in x.

Power Series in x If are constants and x is a variable, then a series of the form is called a power series in x. Some examples are

Radius and Interval of Convergence If a numerical value is substituted for x in a power series then the resulting series of numbers may either converge or diverge. This leads to the problem of determining the set of x-values for which a given power series converges; this is called its convergence set.

Radius and Interval of Convergence If a numerical value is substituted for x in a power series then the resulting series of numbers may either converge or diverge. This leads to the problem of determining the set of x-values for which a given power series converges; this is called its convergence set. Observe that every power series in x converges at x = 0. In some cases, this may be the only number in the convergence set. In other cases the convergence set is some finite or infinite interval containing x = 0.

Radius and Interval of Convergence This leads us to the following theorem.

Radius and Interval of Convergence This theorem states that the convergence set for a power series in x is always an interval centered at x = 0. For this reason, the convergence set of a power series in x is called the interval of convergence.

Radius and Interval of Convergence This theorem states that the convergence set for a power series in x is always an interval centered at x = 0. For this reason, the convergence set of a power series in x is called the interval of convergence. In the case where the convergence set is the single value x = 0 we say that the series has radius of convergence 0.

Radius and Interval of Convergence This theorem states that the convergence set for a power series in x is always an interval centered at x = 0. For this reason, the convergence set of a power series in x is called the interval of convergence. In the case where the convergence set is infinite, we say that it has a radius of convergence of infinity.

Radius and Interval of Convergence This theorem states that the convergence set for a power series in x is always an interval centered at x = 0. For this reason, the convergence set of a power series in x is called the interval of convergence. In the case where the convergence set extends between –R and R we say that the series has radius of convergence R.

Radius and Interval of Convergence Graphically, this is what that looks like.

Finding the Interval of Convergence The usual procedure for finding the interval of convergence of a power series is to apply the ratio test for absolute convergence. We will see that in the following examples.

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d)

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (a)

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (a) This series converges when |x| < 1. At a value of 1, the test is inconclusive. We now need to check the endpoints individually to see if they are included.

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (a) These both diverge, so the interval of convergence is (-1, 1), and the radius of convergence is R = 1.

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (b)

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (b) This ratio is always less than 1, so the series converges absolutely for all values of x. Thus the interval of convergence is and R =

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (c)

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (c) This ratio is never less than 1, so the series diverges for all nonzero values of x. Thus the interval of convergence is x = 0 and R = 0.

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (d)

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (d) This ratio is less than 1 when |x| < 3. Again, the test provides no information when x = +3, so we need to check them separately.

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (d) This is the conditionally convergent harmonic series, so x = 3 is good.

Example 3 Find the interval of convergence and radius of convergence of the following power series. (a) (b) (c) (d) (d) This is the divergent harmonic series, so x = -3 is bad. Interval (-3, 3] and R = 3.

Power Series in x – x 0 If x o is a constant and if x is replaced by x – x o in the power series expansion, then the resulting series has the form This is called a power series in x – x o.

Power Series in x – x 0 If x o is a constant and if x is replaced by x – x o in the power series expansion, then the resulting series has the form This is called a power series in x – x o. Some examples are

Power Series in x – x 0 The first of the previous series is a power series in x – 1 and the second is a power series in x + 3. Note that a power series in x is a power series in x – x o in which x o = 0. More generally, the Taylor Series is a power series in x – x o.

Power Series in x – x 0 The main result on convergence of a power series in x – x o can be obtained by substituting x – x o for x. This leads to the following theorem.

Power Series in x – x 0 It follows from the theorem that the set of values for which a power series in x – x o converges is always an interval centered at x = x o ; we call this the interval of convergence. We can also have convergence only at the point x o and we would say that the series has a radius of convergence of R = 0. The series could also converge everywhere. We say that this has a radius of convergence of infinity.

Power Series in x – x 0 Graphically, this is what that looks like.

Example 4 Find the interval of convergence and radius of convergence of the series

Example 4 Find the interval of convergence and radius of convergence of the series

Example 4 Find the interval of convergence and radius of convergence of the series

Example 4 Find the interval of convergence and radius of convergence of the series Checking the endpoint at 4 by replacing x with 4. This converges absolutely. It also converges by the alternating series test.

Example 4 Find the interval of convergence and radius of convergence of the series Checking the endpoint by replacing x with 6. This is a convergent p-series. The interval of convergence is [4, 6] and the radius of convergence is R = 1.

Homework Section 9.8 Page 667 1, 3, 5, 11, 13, 15, 19, 21, 30 – 48 mult. of 3